This might turn out to be a silly question, but here goes.

Let $\mathcal{C}$ be a full additive subcategory of an abelian category $\mathcal{A}$. I'm wondering if the Grothendieck group $K(\mathcal{C})$ (defined below) depends on $\mathcal{A}$. I can't think of any examples, and I believe that it does not. Something with Yoneda embeddings (see Weibel) comes to mind, but I can't really get it precise.

**Idea**. Given two embeddings $\mathcal{C}\subset\mathcal{A}_1$ and $\mathcal{C}\subset \mathcal{A}_2$, I'm guessing it suffices to find a third one containing these.

**Def**.
Let $\textrm{Ob}(\mathcal{C})$ denote the class of objects in $\mathcal{C}$ and let $\textrm{Ob}(\mathcal{C})/\cong$ be the set of isomorphism classes. Let $F(\mathcal{C})$ be the free abelian group on $\textrm{Ob}(\mathcal{C})/\cong$. To any sequence $$(E) \ \ 0 \longrightarrow M^\prime \longrightarrow M \longrightarrow M^{\prime\prime} \longrightarrow 0 .$$ in $\mathcal{C}$, which is exact in $\mathcal{A}$, we associate the element $Q(E) = [M] - [M^\prime] - [M^{\prime\prime}]$ in $F(\mathcal{C})$. Let $H(\mathcal{C})$ be the subgroup generated by the elements $Q(E)$, where $E$ is a short exact sequence. We define the Grothendieck group, denoted by $K(\mathcal{C})$, as the quotient group $$K(\mathcal{C}) = F(\mathcal{C})/H(\mathcal{C}).$$

(All categories are assumed to be at least skeletally small in this definition, but ok.)